16 problems

Mori structures conjecture. The Mori structures of $U$ are precisely $U$ itself and

Birational invariance conjecture for condition star. Then $X'/\mathbb{P}^1$ also satisfies condition $(\ast)$. This predicts that condition $(\ast)$ is preserved under square-birat…

Semistable dP3 rigidity conjecture. Then $X/\mathbb{P}^1$ is birationally rigid. This is a semistable variant of Grinenko's conjecture.

Grinenko's dP3 rigidity conjecture. If $X$ satisfies condition $(\ast)$, then $X$ is birationally rigid. This conjecture extends the known rigidity theorem under the stronger $K^2$…

Let $U$ be the double space considered in the source, and let ${\mathcal S}$ denote the family of curves used there to construct the fibrations $V_l/\mathbb{P}^1$. Khashin's conjec…

Let $\pi:\mathcal{X}\to B$ be a smooth del Pezzo fibration. Fix an intersection profile $\lambda$, let $N_\lambda$ be the affine subset of $N_1(\mathcal{X})$ consisting of curve cl…

Let $d$ be an integer with $1\leq d\leq 3$. Let $X\to\mathbb P^1$ be a $\mathfrak D$-semistable del Pezzo fibration of degree $d$, and assume that it satisfies the $K$-condition, m…

Dominant-families conjecture. There is some translate $\mathcal{T}$ of $\mathrm{Nef}_{\lambda}$ in $N_{\lambda}$ such that every numerical class in $\mathcal{T}_{\mathbb{Z}}$ is re…

Let $X\to\mathbb{P}^1$ be a del Pezzo fibration of degree $1$ with at most terminal singularities. It satisfies the $K$-condition when $-K_X$ is not in the interior of the movable…

A del Pezzo fibration is birationally rigid if it has only one Mori fiber space structure in its birational class. Let $X$ be a del Pezzo fibration satisfying the hypotheses of The…

Let $X_n$ be the varieties in the series of $G_K$-Mori fiber spaces over $\mathbb{P}^1$ whose general fibers are isomorphic to the degree-$2$ del Pezzo surface $S_K$. Nonrationalit…

Let $G_K\cong \mathrm{PSL}_2(\mathbb{F}_7)$ be the Klein simple group, and let $X_n/\mathbb{P}^1$ be the $G_K$-Mori fiber spaces described above. A $G_K$-Mori fiber space is a thre…

Semistability-modified conjecture. If $X$ satisfies the $K$-condition, then $X$ is birationally rigid.

Let $\mathcal{F}$ be the projective bundle used to construct the threefold Mori dream space $X$, and let $f$ and $g$ be the defining equations of the two hypersurfaces whose comple…

Let $X$ be a standard degree-three del Pezzo fibration, meaning a standard del Pezzo fibration whose generic fibre is a cubic surface. Bi-rigidity conjecture. The threefold $X$ is…

Let $f\colon X\to Z\ni o$ be the germ of a del Pezzo bundle, and let $m_oF_o=f^*(o)$ be a fiber of multiplicity $m_o$, where $m_o\geq 5$ and $F_o$ is of type $I_{2,3,6}$ or…